Sliding Mode Observer

Compute electrical position and mechanical speed of a surface-mount PMSM

Since R2021b

Libraries:
Motor Control Blockset / Sensorless Estimators

Description

The Sliding Mode Observer block computes the electrical position and mechanical speed of a Surface Mount PMSM by using the voltage and current values along the α- and β-axes of the stationary αβ reference frame.

Equations

These equations describe the discrete-time operation of a PMSM:

$i_{α β (k + 1)} = A i_{α β (k)} + B v_{α β (k)} - B e_{α β (k)}$

$e_{α β (k + 1)} = e_{α β (k)} + T_{s} ω_{e (k)} J e_{α β (k)}$

$J = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]$

$Φ = [\begin{matrix} - \frac{R}{L} & 0 \\ 0 & - \frac{R}{L} \end{matrix}]$

$A = e^{Φ T_{s}}$

$B = \int_{0}^{T_{s}} e^{Φ τ} d τ = [\begin{matrix} b & 0 \\ 0 & b \end{matrix}]$

$b = \frac{1 - e^{- R T_{s} / L}}{R}$

These equations describe the discrete-time sliding mode observer operation of a surface mount PMSM:

${\hat{i}}_{α β (k + 1)} = A {\hat{i}}_{α β (k)} + B v_{α β (k)} - B {\hat{e}}_{α β (k)} - η S i g n ({\tilde{i}}_{α β (k)})$

${\hat{e}}_{α β (k + 1)} = {\hat{e}}_{α β (k)} + B^{- 1} g ({\tilde{i}}_{α β (k)} - A {\tilde{i}}_{α β (k - 1)} + η S i g n ({\tilde{i}}_{α β (k - 1)}))$

${\tilde{i}}_{α β (k)} = {\hat{i}}_{α β (k)} - i_{α β (k)}$

${\tilde{e}}_{α β (k)} = {\hat{e}}_{α β (k)} - e_{α β (k)}$

If the back EMF observer fulfils the conditions $| e_{α β (k + 1)} - e_{α β (k)} | \leq m$ and $g \in (0, 1)$ , there exists a k₀, such that:

${\tilde{e}}_{α β (k)} < \frac{m}{g}$

If the sliding mode observer fulfils these conditions:

$g \in (0, 1)$
$| e_{α β (k + 1)} - e_{α β (k)} | \leq m$
$η > b \frac{m}{g}$

then there exists a k=k₀, such that for k≥k₀:

$| {\tilde{i}}_{α β (k)} | \leq η + b \frac{m}{g}$

where:

e_α and i_α are the stator back EMF and current for the α-axis.
e_β and i_β are the stator back EMF and current for the β-axis.
ẽ_α and ĩ_α are the errors in the stator back EMF and current for the α-axis.
ẽ_β and ĩ_β are the errors in the stator back EMF and current for the β-axis.
v_α and v_β are the stator supply voltages.
R is the stator resistance.
L is the stator inductance.
g is the back EMF observer gain.
η is the current observer gain.
ω_e is the electrical angular velocity.
T_s is the sampling period.
k is the sample count.

Tuning Sliding Mode Observer

Use these steps to tune the block using the Current observer gain (η) and Back-emf observer gain (g) parameters.

Select a back-emf observer gain (g) value such that $g \in (0, 1)$ . Bringing g close to the value 1, results in less error in the estimated back-emf. However, this makes convergence slow.
Select a value of m based on the block sample time and maximum slope of the operating back-emf (such that $| e_{α β (k + 1)} - e_{α β (k)} | \leq m$ ).
Select a current observer gain (η) value based on b, m, and g (such that $η > b \frac{m}{g}$ ).

Note

The block functions correctly when you tune the sliding mode observer gains.

When using open-loop control to run a motor, compute the rotor position using both sliding mode observer and an actual sensor hardware and compare the computed position values. If the difference is acceptable, the block functions correctly. Otherwise, manually tune the sliding mode observer gains to ensure that the block functions accurately.

The transition from open-loop control to closed-loop control may fail due to noise in the currents and voltages. To make a successful transition, try reducing the value of the Filter cut-off frequency (Hz) parameter.

Examples

Sensorless Field-Oriented Control of PMSM

Implements the field-oriented control (FOC) technique to control the speed of a three-phase permanent magnet synchronous motor (PMSM). For details about FOC, see Field-Oriented Control (FOC).

Open Model

Ports